Can partial vanishing of Poisson bracket determine local constants of motion?

[giova7_89]

I don't know if this is the right place to post this, but my question is: if i have an Hamiltonian defined on the whole phase space and a function f which is also defined on the whole phase space and doesn't depend explicitly on time, i know that if its poisson bracket with the Hamiltonian vanishes everywhere, f is a constant of the motion. But what happens if this poisson bracket doesn't vanish everywhere, but only on a subset of the phase space? This subset could be for example the one i get from the equation {f,H}=0

Thanks!

 

[Eynstone]

Welcome giova7_89,
If the locus of {f,H}=0 contains a trajectory, f will indeed be constant for that trajectory. If one considers the analytic continuation, one must say that such local constants have little physical significance.